Two ways to fusion energy - ipp.mpg.de · PDF fileheavy ion beam • confinement due ... case that the total energy input used to increase internal energy) ... improvised explosive

Two ways to fusion energy

Inertial fusion:

•  fast heating (e.g. laser) of small pellets, minor explosions •  pressure comparable to solar interior (n~1031 m-3) •  confinement time ~10-10

D-T fusion needs temperatures 10 times larges than in the solar interior: ~150 Mio degrees

Magnetic fusion:

•  confinement through magnetic fields •  very low pressure •  confinement time: a few seconds

Mainly in the US and Japan, keep-in-touch activities in Europe

€

Pfus,ch arg ed = Ploss

€

Pfus,ch arg ednDT2

4⋅ σv

DT⋅ QDT ⋅ 0,2⋅ Vol

€

Ploss ≡3/2⋅ (ne + nDT )⋅ kT

τE⋅ Vol

constTn Ee =⋅⋅ τ constTn Eie =⋅⋅ τ)0(

Reminder: power balance for a fusion device

power gain: power loss :

:Eτ Energy confinement time (characteristic cooling down time)

2~v TDT

σ (at T ~ 10 … 20 keV) Reaction cross section:

Quasi neutrality: ne = nDT

power balance :

Inertial fusion

nτΕ and T are fixed, but pressure p=nT is free to choose Inertial fusion:

•  Fast heating with laser or heavy ion beam •  confinement due to inertia (ion sound wave time scale) •  miniature explosion n large (1031 m-3), τΕ small (10-10 s)

⇒ pressure comparable to the solar core (!)

Estimate required parameters

assume: repetition frequency: ~1 Hz at thermal power of 1GW ⇒ 1 GJ per pellet

for 1GJ energy: fusion of 6x1020 particles necessary (per D-T-pair: 17 MeV=3 10-12J) = 2.4 mg D-T-mix

= pellet with radius of 1.4 mm (ρDT,fl=200 kg/m3)

Required for ignition : confinement time > burning time

Confinement time: ( ) Tk

mRTTk

mRcR

B

i

ieB

iE 23

≈+

==τ (Te=Ti)

Burning time (=time that is needed to burn half of the total fuel):

⎟⎟⎠

⎞⎜⎜⎝

⎛==== unR

unRn

VRnV

DTDTDT

B σσ

τ4

14

4/ 2

Ignition condition for intertial fusion

unTkmR

B

i

σ1

2≥

ρ=mn uTkm

R Bi

σρ

2≥

For T = 10 keV, m=2.5mp, <σu>=2 1022 m3/s : ρR ≥ 30 kg/m2

Required for ignition : confinement time > burning time

⇒  compress pellet to radius of 100µm (same mass ~ 1mg) in about 10-10 s, mass density increases by factor 1000

13

2 34~1~

−

⎟⎠

⎞⎜⎝

⎛ RR

R πρρ

Increase pellet radius? Energy released per pellet increases proportional to volume: 1 GJ ⇒ (150)3 GJ (850 kt TNT)

For pellet with 1mm radius with fluid D-T-mix: ρR = 0.2 kg/m2

Compression methods:

„direct drive“ „indirect drive“

• Ablator is vaporised, repulsion compresses sphere consisting of frozen D-T • Requires very homogeneous exposure to laser!

•  cavity radiation with T of about 100eV •  homogeneous exposure to soft X-ray radiation

http://lasers.llnl.gov/lasers/nif/nif_ife.html#icf

„Hot-Spot“ Concept

Energy balance of compression problematic: •  to achieve required parameters: minimum energy of E=3 NkT (in case that the total energy input used to increase internal energy) •  according to previous requirements : T=10keV, N=6e20 -> E=2.8MJ •  Released energy was 1 GJ, i.e. energy amplification by about 350

But: consider efficiency of the involved processes : -  thermal energy -> elektrical energy (30%) -  efficiency of the laser (10%) -  Transformation of radiation energy in energy of the pellet (5%) for Hohlraum

Total efficiency: 0.3x0.1x0.05=0.0015

„Hot-Spot-concept“

In order to save compressional energy: first, heat only center to 10keV, then, after ignition α-particles heat the rest of the pellet

Compression

Achievable density depends on: a) pressure due to external 'drive’ ( driver energy, ablated material) b) Resistance of target material (entropy, equations of state) c) Hydrodynamic instabilities during implosion (Rayleigh Taylor)

Absorption

Driver

Plasma Korona

Accellaration Compression Burning

Requirements for uniform irradiation

‚Non-uniformity‘ needs to be smaller than ~ 1% RMS (root mean square) Direct drive: For perfect parabolic radiation profile at least 60 laser beams needed Indirect drive: •  Capsule in Hohlraum – radiation field of a black body •  Non-uniformity at small wave lengths significantly reduced. •  Large structure still a problem, e.g.: openings for laser beams

Hydrodynamic Instabilities:

Very homogeneous exposure of the pellet necessary Because: Rayleigh-Taylor instability!

gpdtvd

ρρ −−∇=

Hydrodynamic Equations:

Linearize and v0=0:

00 =∂

∂

tρ

000 =−∇− zgp ρ

1. order:

0011 =∇+

∂

∂ρ

ρ vt zgp

tv

111

0 ρρ −−∇=∂

∂ 01 =⋅∇ v

( ) 0=⋅∇+∂

∂ vt

ρ

ρ

Continuity equation:

zgpvvtv

ρρ −−∇=⎟⎠

⎞⎜⎝

⎛ ∇⋅+∂

∂

Forces:

0=⋅∇ v

Incompressibily:

ρ0 varies only in z direction:

ansatz: ( ) ( )( )tykxkizXX yx γ++= exp11

001

1 =∂

∂+

∂

∂

zv

t zρρ

Continuity:

0110 =

∂

∂+

∂

∂

xp

tv xρ 011

0 =∂

∂+

∂

∂

yp

tv yρ 01

110 =+

∂

∂+

∂

∂ gzp

tv z ρρforces:

0111 =∂

∂+

∂

∂+

∂

∂

zv

yv

xv zyxIncompressibity:

0110 =

∂

∂+

∂

∂

xp

tv xρ 011

0 =∂

∂+

∂

∂

yp

tv yρ 01

110 =+

∂

∂+

∂

∂ gzp

tv z ρρ

forces:

0110 =+ pikv xxργ 0110 =+ pikv yyργ

0

11 ργ

pikv xx −=

Incompressibility: 0111 =

∂

∂++

zvvikvik z

yyxx

01

0

12

0

12

=∂

∂++

zvpkpk zyx

ργργ

zv

kp z

∂

∂−= 1

20

1ργ

ansatz: ( ) ( )( )tykxkizXX yx γ++= exp11

gvzv

zkzp

zz

1101

021 ργρρ

γ−−=⎟

⎠

⎞⎜⎝

⎛∂

∂

∂

∂−=

∂

∂

Forces in z direction:

zv

kp z

∂

∂−= 1

20

1ργ01

110 =+

∂

∂+

∂

∂ gzp

tv z ρρ

zzz v

zgkvk

zv

z 10

2

2

1021

0 ∂

∂−=⎟

⎠

⎞⎜⎝

⎛∂

∂

∂

∂ ργ

ρρ

zv z ∂

∂−= 0

11ρ

ργFrom continuity:

zz vk

zg

zv

z 1020

02

10

11 ρρ

ργρ ⎥

⎦

⎤⎢⎣

⎡

∂

∂−=⎟

⎠

⎞⎜⎝

⎛∂

∂

∂

∂

Boundary condition for ∞→z

0,0 11 =

∂

∂=

zvv z

z

zz vk

zg

zv

z 1020

02

10

11 ρρ

ργρ ⎥

⎦

⎤⎢⎣

⎡

∂

∂−=⎟

⎠

⎞⎜⎝

⎛∂

∂

∂

∂

∞→z

0,0 11 =

∂

∂=

zvv z

z

For z≠0: 00 =∂

∂

zρ

zz vk

zv

12

21

2

=∂

∂

kzkzz ezezzv −Θ+−Θ )()(~)(1General solution:

Integration , left: ∫−

ε

ε

dz...ε

ε

ε

ε

ρρ−− ∂

∂=⎟

⎠

⎞⎜⎝

⎛∂

∂

∂

∂∫ z

vdzzv

zzz 1

01

0

( ) ( ) ( ) ( )[ ] ( )zvzkezkezezezv

zkzkzkzkzz

11 Θ−−Θ+−−=∂

∂ −− δδ

∂∂z

ρ0∂v1z∂z

!

"#

$

%&

−ε

ε

∫ dz = ρ0 ε( ) k Θ −ε( )− k Θ ε( )( )− ρ0 −ε( ) k Θ ε( )− k Θ −ε( )( )!" #$v1z 0( )

= −k ρ0 ε( )+ ρ0 −ε( )!" #$v1z 0( )

Boundary condition for

zz vk

zg

zv

z 1020

02

10

11 ρρ

ργρ ⎥

⎦

⎤⎢⎣

⎡

∂

∂−=⎟

⎠

⎞⎜⎝

⎛∂

∂

∂

∂

Integration , right: ∫−

ε

ε

dz...

00

10

0

110 00=+= ∫∫∫

−

−

−

dzvdzvdzv zzz

ε

ε

ε

ε

ρρρ

[ ] ( )012

2120

2 00 zz vkgdzvkz

g −+

−

−−=∂

∂− ∫ ρρ

γρ

γ

ε

ε

∂∂z

ρ0∂v1z∂z

!

"#

$

%&

−ε

ε

∫ dz = −k ρ0 ε( )+ ρ0 −ε( )!" #$v1z 0( )

zz vk

zg

zv

z 1020

02

10

11 ρρ

ργρ ⎥

⎦

⎤⎢⎣

⎡

∂

∂−=⎟

⎠

⎞⎜⎝

⎛∂

∂

∂

∂

∞→z

0,0 11 =

∂

∂=

zvv z

z

Integration results in: ∫−

ε

ε

dz... −k ρ0+ + ρ0

−"# $%v1z (0) = −gk2

γ 2ρ0+ − ρ0

−"# $%v1z (0)

−+

−+

+

−≡=

00

002 ,ρρρρ

γ AAkg A: Atwood-Zahl

Boundary condition for

Unstable for heavier fluid above the lighter one

∂∂z

ρ0∂v1z∂z

!

"#

$

%&

−ε

ε

∫ dz = −k ρ0 ε( )+ ρ0 −ε( )!" #$v1z 0( )

[ ] ( )012

2120

2 00 zz vkgdzvkz

g −+

−

−−=∂

∂− ∫ ρρ

γρ

γ

ε

ε

Cold, dense liquid is accelerated trough hot, less dense liquid

Rayleigh-Taylor-Instability

Growth rate of the RT instability driven by ablation is smaller than the one of the classical RT instability

As derived above: for classical RT instability

Linear regime:

Non-linear regime: Ascending bubbles, Falling spikes.

Atwood Zahl. A =−

+

ρ ρρ ρ

''

Wachstumsrate Akg=γ

amplitude a t a e t( ) = ⋅ ⋅0

γAblative driven RT: ‘Takabe formula’

akvkg 39.0 −=γ

Reduction in growth rate due to ablation of “perturbed” material and due to finite width of the affected region

graphics from http://www.llnl.gov/asci/gallery/

Rayleigh-Taylor Instability

Pellet shows perturbations with medium-size wave numbers

Estimate: pellet with high Z coating (0.1mm): 10g/cm3

Ablation pressure: 100Mbar à acceleration (~dp/rho) 1013m/s

For wave number equal to coating thickness: inverse growth rate ~ 10-9s = similar to confinement time! Rayleigh-Taylor instability cannot be avoided additionally, inhomogeneity has to be less than 1%

Fast Ignition

Examples for parameter reached so far:

Density of compressed target: 1000 g/cm3 (Osaka,Japan) Plasma temperature : > 10 keV several labs But not simultaneously! Example : NOVA-Laser (Livermore) nTτE=5 1020 m-3 keV s radiation temperature in cavity: 250 eV Since 2009 : NIF, new laser, 20 times more power than NOVA 2012: ICF program officially terminated, focus shift to material sciences 2013: new results with improved confinement •  pellet hot core heated by fusion with 50%, the other 50% by

compression •  1 % of the energy deposited in the Hohlraum into fusion energy

NIF: National Ignition Facility https://lasers.llnl.gov/

master oszillator: ytterbium-doped optical fiber laser

Amplifier: laser diode-pumped neodymium-doped glass

1.33 megajoules (MJ) of 3ω (ultraviolet) light to the hohlraum with 365 terawatts (TW) of peak power.

Expected energy gain for different concepts

S. Nakai, K. Mima, Rep. Prog. Phys. 2004

required laser energies

S. Nakai, K. Mima, Rep. Prog. Phys. 2004

Speed-of-Light Weapons The tailored-aperture ceramic laser (TACL) and solid-state heat-capacity laser (SSHCL) are examples of speed-of-light directed-energy weapons that can target and destroy short-range rockets, missiles, artillery, mortar fire, unmanned aerial vehicles and other battlefield threats such as improvised explosive devices (IEDs) and landmines. LLNL's work on the SSHCL has set the stage for a new generation of ceramic lasers and high-power laser architectures which will be capable of running continuosly at high efficiency and with exceptional beam quality……

[Source: LLNL web site]